Supplementary Information for:Static and lattice vibrational energy differencesbetween polymorphs

Jonas Nyman and Graeme M. Day

1 Additional results

0 2 4 6 8 10 12 14∆ Lattice energy [%]

0

20

40

60

80

100

120

140

Num

bero

fpai

rs

Figure S1: The distribution of differences in relative lattice energy.

0 2 4 6 8 10 12 14∆ Lattice energy [kJ/mol]

0

10

20

30

40

50

60

70

Num

bero

fpai

rs

Figure S2: The distribution of lattice energy differences for structures with hydrogen bonding.

0 2 4 6 8 10 12 14∆ Lattice energy [kJ/mol]

05

1015202530354045

Num

bero

fpai

rs

Figure S3: The distribution of lattice energy differences for structures without hydrogenbonding.

1

Electronic Supplementary Material (ESI) for CrystEngComm.This journal is © The Royal Society of Chemistry 2015

0 2 4 6 8 10 12 14∆ Helmholtz energy [kJ/mol]

0

10

20

30

40

50

60

70

80

Num

bero

fpai

rsFigure S4: The distribution of free energy differences for structures with hydrogen bonding.

0 2 4 6 8 10 12 14∆ Helmholtz energy [kJ/mol]

05

1015202530354045

Num

bero

fpai

rs

Figure S5: The distribution of free energy differences for structures without hydrogen bonding.

0 2 4 6 8 10 12 14∆ Entropy [J/mol·K]

0

10

20

30

40

50

60

70

Num

bero

fpai

rs

Figure S6: The distribution of entropy differences for structures with hydrogen bonding.

0 2 4 6 8 10 12 14∆ Entropy [J/mol·K]

0

5

10

15

20

25

30

35

Num

bero

fpai

rs

Figure S7: The distribution of entropy differences for structures without hydrogen bonding.

2

0.0 0.2 0.4 0.6 0.8 1.0∆Cv [J/mol·K]

0

10

20

30

40

50

60

70

Num

bero

fpai

rsFigure S8: The distribution of heat capacity differences for structures with hydrogen bonding.

0.0 0.2 0.4 0.6 0.8 1.0∆Cv [J/mol·K]

0

10

20

30

40

50

Num

bero

fpai

rs

Figure S9: The distribution of heat capacity differences for structures without hydrogenbonding.

0 2 4 6 8 10∆ Density [%]

0

10

20

30

40

50

60

70

80

Num

bero

fpai

rs

Figure S10: The distribution of density differences for structures with hydrogen bonding.

0 2 4 6 8 10∆ Density [%]

0

10

20

30

40

50

Num

bero

fpai

rs

Figure S11: The distribution of density differences for structures without hydrogen bonding.

Since we have not studied conformational polymorphs, we expect the in-tramolecular energy differences to be small, see Figure S12. A few polymorphs

3

0 5 10 15 20∆ Intramolecular energy [kJ/mol]

0

50

100

150

200

250

300

350

400

Num

bero

fpai

rsFigure S12: The distribution of intramolecular energies.

do however exhibit different inter- and intramolecular hydrogen bonding motifs,so that intramolecular energy differences can be as large as 19.4 kJ/mol. TheCSD structure codes of the outliers, and their calculated intramolecular en-ergy differences, are: HDXMOR/HDXMOR01 (∆Eintra = 14.4 kJ/mol); LEZ-JAB/LEZJAB01 (∆Eintra = 15.6 kJ/mol); DMANTL01/DMANTL07 (∆Eintra =15.7 kJ/mol) and IFOVOO/IFOVOO01 (∆Eintra = 19.4 kJ/mol).

The next highest ∆Eintra are: BOHZOO/BOHZOO01 (∆Eintra = 13.3 kJ/mol);FUGJUM/FUGJUM01 (∆Eintra = 11.9 kJ/mol); PABZAM/PABZAM01 (∆Eintra =11.4 kJ/mol) and FEGWAP/FEGWAP01 (∆Eintra = 11.2 kJ/mol).

4

2 Selected polymorph pairs

An alphabetically sorted list of CSD reference codes for all polymorphs includedin this study can be found in polymorphs.txt

3 Distribution of molecular RMSD in polymorphs

The molecular conformations tends to be very similar in polymorphs. Con-formational polymorphism was studied by Cruz-Cabeza and Bernstein 1. We

Figure S13: Molecular RMSD between 1397 single-component polymorph pairs.

observe the same distribution of molecular RMSD as did Cruz-Cabeza andBernstein1. We have used a limit of 0.25 Å to remove conformational poly-morphs.

5

4 Co-prime splitting of linear supercells.

The crystal unit cell is expanded into linear supercells (1x1xn) where n is thenumber necessary to reach the target k-point distance. If n < 5 the latticedynamic calculations is performed on the supercell as is. For n > 5 the supercellis split into several smaller supercells (1x1xk, 1x1x`, 1x1xm ...) such thatk, `,m... are all mutually co-prime and k + ` + m... > n. The long linearsupercells are split into 2, 3 or 4 co-prime supercells according to the scheme inTable S1. Note that this is by no means the only possible choice, and we make noclaim that this is the best or computationally most efficient splitting. Splittingthe supercells means that the sampled k-points will no longer be equidistantlyplaced along the reciprocal axes, but this should have a negligible effect onthe results. Unfortunately though, it also means that the convergence will notbe monotonic with respect to the target k-point sampling distance, makingconvergence testing difficult.

6

Table S1: Linear supercells were split into 2, 3 or 4 smaller supercells with mutually co-primeexpansion coefficients in this way.

2 → 23 → 34 → 45 → 2, 36 → 3, 47 → 3, 48 → 3, 59 → 2, 3, 510 → 2, 3, 511 → 3, 4, 512 → 3, 4, 513 → 3, 4, 714 → 3, 4, 715 → 3, 5, 716 → 4, 5, 717 → 2, 3, 5, 718 → 5, 6, 719 → 3, 4, 5, 720 → 4, 7, 921 → 5, 7, 922 → 5, 8, 923 → 2, 5, 7, 924 → 7, 8, 925 → 5, 9, 1126 → 7, 9, 10

7

5 Revised Williams99 parameters

All parameters describing interactions between C, N, O and H atoms are de-scribed using Williams’ original Williams99 forcefield parameters, apart fromhydrogen bond H...A interactions, which were reparameterised to work moreeffectively with the atomic multipole electrostatic model. For H...A interac-tions, the pre-exponential parameter of the exp-6 model was modified from theWilliams99 value. The parameters are given in the following table.

Table S2: Revised H...A parameters in the exp-6 intermolecular model used. C = 0 for allinteractions.

hydrogen acceptor A (eV)H2 N1 149H2 N2 166H2 N3 163H2 O1 129H2 O2 105H3 N1 70H3 N2 118H3 O1 127H3 O2 133H4 N1 141H4 N2 77H4 N3 56H4 N4 112H4 O1 34H4 O2 198

8

6 Model potential parameters for halogens

Halogen atoms tend to have an anisotropic van der Waals radius2. To accountfor this, intermolecular potentials with an anisotropic repulsion term has beendeveloped3,4. A local unit vector ez is defined at each anisotropic site, parallelto the covalent bond joining the halogen to its bonded atom, pointing away fromthe bond. A second unit vector, eik, is the vector between the interacting atoms.DMACRYS describes repulsion anisotropy using a modified exp-6 potential ofthe form:

V = G exp (−Bικ(rij − ρικ(Ωik)))− Cικ/r6, (1)where ρικ(Ωik) describes the anisotropy of repulsion, and is defined as:

ρικ(Ωik) = ρικ0 +ρ

ι1(e

iz ·eik)+ρκ1(−ekz ·eik)+ρι2(3[eiz ·eik]2−1)/2+ρκ2(3[ekz ·eik]2−1)/2

(2)ρ0 describes the isotropic repulsion, ρ1 parameters describe a shift of the

centre of repulsion and ρ2 parameters describe a quadrupolar distortion of theatom. Parameters for Cl and F were taken from Day’s specifically developedpotential for molecule XIII in the 4th blind test of crystal structure prediction5.Halogen parameters were empirically fitted to reproduce the crystal structuresof a set of halogenated aromatic molecules. Details are available in the ESI tothe 4th blind test paper. The parameters, in input format for DMACRYS areprovided below:

BUCK F_01 F_01

3761.006673 0.240385 7.144500 0.0 70.0

ANIS F_01 F_01

0 0 2 0 2 -0.035000

0 0 0 2 2 -0.035000

ENDS

BUCK Cl01 Cl01

5903.747391 0.299155 86.716330 0.0 70.0

ANIS Cl01 Cl01

0 0 2 0 2 -0.093860

0 0 0 2 2 -0.093860

ENDS

9

7 Additional convergence plots

Convergence of free energy, heat capacity, zero point energy and entropy withrespect to k-point sampling and the number of k-points was studied. Since wein this article are only interested in the difference between crystal structures,we do not show results for the convergence of the “absolute” quantities, butsuch convergence tests have been made previously6. Here we show two typicalresults from our convergence testing.

0 5 10 15 20 25Number of sampled k-points

80

85

90

95

100

Ent

ropy

[J/m

ol·K

]

T = 300.00 K

Figure S14: Convergence of the entropy for a crystal structure of theophylline with respect tothe number of sampled k-points. Plot reproduced from Nyman 6

0.00.20.40.60.81.0

Target k-point distance [Å−1]

80

85

90

95

100

Ent

ropy

[J/m

ol·K

]

T = 300.00 K

Figure S15: The same data as in the diagram above, but now over the k-point distance. Plotreproduced from Nyman 6

To make the “absolute” thermodynamic quantities converge is difficult andrequire a careful k-point sampling. With our co-prime split linear supercell

10

sampling, the convergence is no longer monotonic. However, the difference inentropy, vibrational energy etc between polymorphs converges slightly faster.

0.00.20.40.60.81.01.21.4

Target k-point distance [Å−1]

−3

−2

−1

0

1

2

3

∆∆A

[kJ/

mol

]

XULDUD-XULDUD01MALEHY01-MALEHY10MALEHY12-MALEHY01II-IIV-IVI-IVII-I

Figure S16: Convergence of ∆A for a set of theophylline (Roman numerals), maleic hydrazide,and 3,4-cyclobutylfuran polymorphs with respect to k-point distance.

We have estimated the error in ∆S at 300 K due to the k-point samplingto ±1 J/mol·K at a k-point distance of 0.12 Å. The error in Helmholtz energycaused by the incomplete k-point sampling at the same temperature is about1 kJ/mol, probably smaller than the error in lattice energy. The zero pointenergy and heat capacity converges much more rapidly. The errors for thesequantities are negligible.

Errors arise both because of the finite sampling, and the selective way ofsampling along only three directions. The error caused by neglecting the “diag-onal” k-vectors is however small and likely systematic, not affecting the delta-quantities significantly.

The thermodynamic properties are functions of the phonon density of states,and an alternative way to judge the convergence is to compare phonon DOScalculated at different k-point samplings. In the article, we show the phonondensity of states for two theophylline and two maleic hydrazide polymorphscalculated at a k-point sampling of 0.12 Å−1. Below, we show the density ofstates of the same structures, but at a denser target k-point distance of 0.08Å−1. The density of states calculated at 0.12 and 0.08 Å−1 are very similar andwell converged at these samplings.

11

0 50 100 150 2000.000

0.005

0.010

0.015

0.020BAPLOT04

0 50 100 150 200Phonon frequency [cm−1]

0.000

0.005

0.010

0.015

0.020BAPLOT06

Figure S17: Phonon density of states for theophyline polymorphs I and II calculated with atarget k-point distance of 0.08 Å−1.

0 50 100 150 2000.000

0.005

0.010

0.015

0.020MALEHY01

0 50 100 150 200Phonon frequency [cm−1]

0.000

0.005

0.010

0.015

0.020MALEHY12

Figure S18: Phonon density of states for two maleic hydrazide polymorphs calculated with atarget k-point distance of 0.08 Å−1.

12

References

[1] A. J. Cruz-Cabeza and J. Bernstein, Chemical reviews, 2013, 114, 2170–2191.

[2] S. C. Nyburg and C. H. Faerman, Acta Crystallographica Section B, 1985,41, 274–279.

[3] G. M. Day and S. L. Price, Journal of the American Chemical Society, 2003,125, 16434–16443.

[4] S. L. Price, M. Leslie, G. W. A. Welch, M. Habgood, L. S. Price, P. G.Karamertzanis and G. M. Day, Phys. Chem. Chem. Phys., 2010, 12, 8478–8490.

[5] G. M. Day, T. G. Cooper, A. J. Cruz-Cabeza, K. E. Hejczyk, H. L. Ammon,S. X. M. Boerrigter, J. S. Tan, R. G. Della Valle, E. Venuti, J. Jose, S. R.Gadre, G. R. Desiraju, T. S. Thakur, B. P. van Eijck, J. C. Facelli, V. E.Bazterra, M. B. Ferraro, D. W. M. Hofmann, M. A. Neumann, F. J. J.Leusen, J. Kendrick, S. L. Price, A. J. Misquitta, P. G. Karamertzanis,G. W. A. Welch, H. A. Scheraga, Y. A. Arnautova, M. U. Schmidt, J. van deStreek, A. K. Wolf and B. Schweizer, Acta Crystallographica Section B, 2009,65, 107–125.

[6] J. Nyman, In Silico Predictions of Porous Molecular Crystals and Clathrates,2014, M.Phil thesis, University of Southampton.

13